Abstract:
A sufficient condition for the absence of tangent transformations admitted by second-order quasi-linear differential equations and a sufficient condition for linear autonomy of operators of the Lie group of transformations admitted by second-order weakly nonlinear differential equations are found. A theorem on the structure of the first-order conservation laws for second-order weakly nonlinear differential equations is proved. A classification of second-order linear differential equations with two independent variables in terms of first-order conservation laws is proposed.
\Bibitem{Chi09}
\by Yu.~A.~Chirkunov
\paper On group properties and conservation laws for second-order quasi-linear differential equations
\jour Prikl. Mekh. Tekh. Fiz.
\yr 2009
\vol 50
\issue 3
\pages 64--70
\mathnet{http://mi.mathnet.ru/pmtf1741}
\elib{https://elibrary.ru/item.asp?id=11928582}
\transl
\jour J. Appl. Mech. Tech. Phys.
\yr 2009
\vol 50
\issue 3
\pages 413--418
\crossref{https://doi.org/10.1007/s10808-009-0055-5}
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https://www.mathnet.ru/eng/pmtf1741
https://www.mathnet.ru/eng/pmtf/v50/i3/p64
This publication is cited in the following 7 articles:
Yu. A. Chirkunov, “Description of the attenuation of invariant ultrasonic beams after formation of the shock fronts in a cubic nonlinear medium in the absence of dissipation”, Communications in Nonlinear Science and Numerical Simulation, 117 (2023), 106942
Yu. A. Chirkunov, “Invariant submodels describing a propagation of the ultrasonic beams in a cubically nonlinear medium without dissipation after self-focusing”, International Journal of Non-Linear Mechanics, 133 (2021), 103731
Yu.A. Chirkunov, E.O. Pikmullina, “Invariant submodels of the model of thermal motion of gas in a rarefied space”, International Journal of Non-Linear Mechanics, 95 (2017), 185
Yu A Chirkunov, “Motion of gas in highly rarefied space”, J. Phys.: Conf. Ser., 894 (2017), 012108
Yu.A. Chirkunov, “Exact solutions of the system of the equations of thermal motion of gas in the rarefied space”, International Journal of Non-Linear Mechanics, 83 (2016), 9
Yu. A. Chirkunov, “Invariant solutions of the Westervelt model of nonlinear hydroacoustics without dissipation”, International Journal of Non-Linear Mechanics, 85 (2016), 41
Yu.A. Chirkunov, “Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin's model”, Journal of Applied Mathematics and Mechanics, 79:5 (2015), 506
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